Modeling emission-generating technologies: reconciliation of axiomatic and by-production approaches

See also the general equilibrium analysis in Murty (2010), which distinguishes between the technologies of pollution-generating and victim firms. The latter possess Starrett-type non-convexities, while the former satisfy costly disposal assumptions similar to the ones discussed in MRL and Murty (2015a). Murty also provides diagrammatic and numerical examples of such non-convexities when a firm’s emissions may prove harmful for its own intended production.

For the general case that also allows emission generation by (some) intended outputs, see Murty (2015a).

The relation \(=:\) means that the argument on the right is defined by the argument on the left.

In line with material-balance conditions, the US-Energy Information Agency and EPA reports (2009, 2014) estimate uncontrolled (gross) emissions from fossil fuel combustion by multiplying fuel-specific emission factor, which is based on emission-causing content such as the sulfur content of the respective fuel, by fuel consumption and by boiler firing configurations. Though \(\hbox {CO}_2\) control technologies that can be installed in fossil-fueled electricity generating plants are in the early stages of research, environmental regulation requires fossil-fueled electricity generating plants to install pollution abatement equipments such as flue gas desulfurization (FGD) units, low NO\(_x\) burners, selective catalytic reduction systems, etc. See also Moslener and Requate (2007). Controlled (net) emissions are then computed by taking into account efficiency of the plants’ FGD units or reduction percentages. For example, data reveal (see Srivastava and Jozewicz 2001) that advanced wet scrubbers (FGD technology) can provide \(\hbox {SO}_2\) reductions in excess of 95 %. At national or levels, a source of carbon sequestration (\(\hbox {CO}_2\) abatement) is provided by forests. Annual change in the stock of forests is a measure of carbon capture by forests during the year. These figures are provided by Global Forest Resources Assessment (FRA) (see, e.g., FRA 2008) and are used to estimate net \(\hbox {CO}_2\) emissions by countries (see, e.g., Intergovernmental Panel on Climate Change (IPCC) (1996) and Murty (2015b). Thus, in theory, the minimal amounts of emissions generated could be zero if sufficient cleaning-up operations are performed.

For example, fuels such as natural gas and petrol contain hydrocarbons. Combustion of these fuels can be complete or incomplete. Combustion is complete when there is enough supply of oxygen, so that carbon oxidizes completely to carbon dioxide (\(\hbox {CO}_2\)) and hydrogen oxidizes to water. When oxygen supply is limited, then combustion is incomplete and more of carbon monoxide (CO) along with soot (carbon) is produced rather than \(\hbox {CO}_2\), and it is possible that some of the hydrogen in the fuel remains unreacted. Thus, the extents of \(\hbox {CO}_2\) and water produced as by-products of combustion depend on the supply of oxygen available during combustion. In industrial applications and in fires, air is the source of oxygen. In air, oxygen is mixed with nitrogen. Nitrogen does not take part in combustion, but at high temperatures, some of it will be converted to NO\(_x\). Thus, the extent of NO\(_x\) generated during combustion depends on the temperature level at which combustion is conducted (see EPA 1999).

To draw parallels with standard (non-emission generating) technologies, note that these technologies also contain both lower and upper boundaries on intended outputs. However, the (analytically uninteresting) lower boundary of zero outputs is automatically imposed by the restriction of the technology to the nonnegative orthant.

Vector notation: \(\bar{x}\ge x\) if \(\bar{x}_i\ge x_i\) for all \(i; \bar{x}> x\) if \(\bar{x}_i\ge x_i\) for all i and \(\bar{x}\ne x\); and \(\bar{x}\gg x\) if \(\bar{x}_i> x_i\) for all i.

See Murty (2015a) for analysis of the general case where emissions generated by a producing unit can also have detrimental or beneficial affects on its own intended-output production possibility set.

Recall, from part (iii) of Remark 4, it is not automatically guaranteed that \(T^y(x_z, x_o, a,\bar{z})\ne \emptyset \). The material-balance component of the given emission-generating technology might not permit input vector \(\langle x_z, a\rangle \) to produce emission vector \(\bar{z}\). For example, if \(\bar{z}<z\), it is possible that the given levels of emission-causing inputs \(x_z\) may be too large and the given level of abatement a may be too low to generate \(\bar{z}\), so that \(T^y(x_z, x_o, a,\bar{z})=\emptyset \).

Recall, from part (ii) of Remark 4 that it does not automatically follow that \(T^z(x_z, \bar{x}_o, a, \bar{y})\ne \emptyset \). For example, if levels of non-emission causing inputs fall a lot or if intended-output production is increased significantly then, given the resources, the original vector of cleaning-up output levels may no longer be feasible.

Recall that part (ii) of Remark 4 says it is possible that, when emission-causing inputs are decreased or the cleaning-up outputs are increased, the current levels of intended outputs may no longer be technologically feasible. Hence, (5.6) says that decreases in emission-causing inputs or increases in cleaning-up outputs can continue producing the same levels of emissions if and only if such changes can continue producing the existing levels of intended outputs.

Single-equation modeling of pollution-generating technologies, following the lead of Baumol and Oates (1975, 1988), was the principal mainstream approach for years. This approach simply treats pollution as just another input in a single production relationship. The principal alternative (employing mathematical programming methods) has been the “weak disposability” approach first proposed by Färe and Grosskopf (1983). As shown by Førsund (2009) and MRL, this approach also has counterintuitive implications for trade-offs in the production process.

See Blackorby et al. (1978), Färe and Primont (1995), and Russell (1998) for extensive discussions of radial distance functions.

Examples of the these types of technologies are discussed in Sect. 9.

Note an error in Murty (2015a), where a similar distance function \(D_2\) is stated to be homogeneous of degree 1 in z.

Murty (2015a) formulates both distance functions in the subspace of intended outputs and emissions. Formulated thus, the representation theorem in her paper requires, in addition to assumptions (EG0), (EG1), and (EG2), the assumption that the set \(T^{y,z}(x,a)\), when not empty, is convex. No additional convexity assumption is required when the two distance functions are formulated, as in the current paper, in two different spaces: \(D^{\mathrm{EG}}_1\) [as defined in (6.1)] represents the upper frontier of intended outputs and is formulated in the space of intended outputs, while \(D^{\mathrm{EG}}_2\) [as defined in (6.2)] represents the lower frontier of emissions and is formulated in the space of emissions. The following representation theorem eschews Murty’s assumption of convexity of the technology.

These trade-offs can be computed using the implicit function theorem when the two distance functions are differentiable. See Sect. 9 for some examples.

See also footnote 5.

\(\hbox {SO}_2\) and \(\hbox {CO}_2\) emission factors of different types of coal based on material-balance considerations (such as the sulfur and carbon contents of a given type of coal) and environmental efficiencies in combustion are provided by US-Energy Information Agency and EPA reports.

See also footnote 5. Note that the parameter restrictions assure that the requisite homogeneity and monotonicity conditions are satisfied.

Apologies for the abuse of notation (in the interest of simplicity).

Note again that this function satisfies the requisite homogeneity and monotonicity properties. Note also that the ray through the cusps of the level sets for different values of input and abatement output vectors is given by (\(z_1=(\alpha _1/\alpha _2)z_2\)).

See Coelli et al. (2007), Murty et al. (2012) and Serra et al. (2014) for alternative approaches to the measurement of environmental efficiency.

We are currently at work on a handbook chapter that compares and endeavors to synthesize these models.